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Yang--Baxter maps (YB maps) are set-theoretical solutions to the quantumYang--Baxter equation. For a set $X=\Omega\times V$, where $V$ is a vectorspace and $\Omega$ is regarded as a space of parameters, a linear parametric YBmap is a YB map $Y\colon X\times X\to X\times X$ such that $Y$ is linear withrespect to $V$ and one has $\pi Y=\pi$ for the projection $\pi\colon X\timesX\to\Omega\times\Omega$. These conditions are equivalent to certain nonlinearalgebraic relations for the components of $Y$. Such a map $Y$ may be nonlinearwith respect to parameters from $\Omega$.  We present general results on such maps, including clarification of thestructure of the algebraic relations that define them and severaltransformations which allow one to obtain new such maps from known ones. Also,methods for constructing such maps are described. In particular, developing anidea from [Konstantinou-Rizos S and Mikhailov A V 2013 J. Phys. A: Math. Theor.46 425201], we demonstrate how to obtain linear parametric YB maps fromnonlinear Darboux transformations of some Lax operators using linearapproximations of matrix refactorisation problems corresponding to Darbouxmatrices. New linear parametric YB maps with nonlinear dependence on parametersare presented.

Yang--Baxter maps, Darboux transformations, and linear approximations of refactorisation problems